ERIC EJ1114045: Why Are Mathematics Teachers "Not Sure"?

2016·0.48 MB·English

by  ERIC

Mathematics Teacher Education and Development 2016, Vol 18.2, 107-123
Why are mathematics teachers “not sure”?1

Janne Fauskanger Reidar Mosvold
University of Stavanger University of Stavanger
Received: 25 January 2016 Accepted: 11 July 2016
© Mathematics Education Research Group of Australasia, Inc.
Researchers have widely adopted measures of teachers’ mathematical knowledge for teaching
(MKT). This paper investigates why teachers select “I’m not sure” as a suggested solution in MKT
items. In this study, in-service teachers responded to multiple-choice MKT items, they submitted
written responses to open-ended questions, and they discussed these items in group discussions. We
analyse the different kinds of responses from three teachers in depth. A framework of cognitive types
of teacher knowledge—distinguishing between Type 1, Type 2 and Type 3 knowledge—was applied
in the analysis. We suggest that one teacher selected “I’m not sure” due to insufficient MKT, whereas
the other two teachers appeared to have Type 1 and Type 2 knowledge. These findings provide proof
that the knowledge teachers utilise in responses to open-ended questions and group discussions does
not necessarily mirror the knowledge used when selecting a particular multiple-choice response.
Keywords . teachers’ mathematical knowledge for teaching . measurement . multiple-choice
items
Introduction
This paper builds upon the practice-based theory of mathematical knowledge for teaching
(MKT), as it was developed by a group of researchers at the University of Michigan directed by
Deborah Ball (e.g. Ball, Thames & Phelps, 2008). Associated with their development of the
practice-based theory of MKT, the researchers in Michigan designed multiple-choice items to
measure MKT (e.g. Hill, Sleep, Lewis & Ball, 2007). The item design was mainly carried out as
part of the Learning Mathematics for Teaching (LMT) project, and the LMT project continues to
administer and provide training in the use of these items—often referred to by researchers as the
MKT items or MKT measures (e.g. Fauskanger, Jakobsen, Mosvold & Bjuland, 2012). Based on
previous classroom studies, researchers in the LMT project developed items that focused on the
mathematical knowledge needed for teaching (Ball et al., 2008). The MKT measures include items
that reflect the mathematical tasks teachers face in classrooms. One example is assessing student
work. According to Hill et al. (2007) assessments composed of these tasks of teaching can be used
to measure the effectiveness of mathematics-focused professional development.
Developing assessment items that measure teachers’ knowledge is complicated and
expensive, and only a few standardised instruments for measuring the knowledge needed for

1 This paper is developed from a paper that was originally presented at the Thematic Working Group (TWG) 20 on
“Mathematics teacher knowledge, beliefs and identity” at CERME9 in Prague, Czech Republic, February 2015 (see
Fauskanger & Mosvold, 2015c).

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Why are mathematics teachers “not sure”? Fauskanger & Mosvold
teaching mathematics internationally exist. In addition, the knowledge requirements of teaching
are highly debated, and it is thus far from straightforward to talk about knowledge “needed” for
teaching mathematics (e.g. Blömeke & Delaney, 2012). Among the instruments that aim at
measuring such knowledge, however, the MKT measures are among the most widely adopted
(Blömeke & Delaney, 2012). The MKT measures seem to function well psychometrically across
countries (e.g. Blömeke & Delaney, 2012; Fauskanger et al., 2012). Like with any measures, there
have also been challenges with the MKT measures. For instance, particular item stems and/or
item distracters seem to be problematic (e.g. too difficult, difficult to understand) and display
differences in item difficulty or negative point-biserial correlation; this has been observed in
different countries (e.g. Ng, Mosvold & Fauskanger, 2012). It has also been observed that minor
modifications of items might influence teacher performance (e.g., Kwon, Thames & Pang, 2012).
In the context of writing multiple-choice items to measure teachers’ specialised knowledge for
teaching mathematics, Orrill et al. (2015) have identified five closely interrelated challenges. The
purpose of this study is to dig deeper into one of these challenges: “writing clear stems and
distracters” (p. 17).
A standard multiple-choice item includes a correct answer—referred to as the key—and a
selection of distracters. The distracters are always coded as wrong answers. In the LMT project,
some items were formatted with “yes” and “no” as alternative solutions (see Fig. 1). From a test-
theoretical perspective, such items might invite test-takers to guess. To avoid this, “I’m not sure”
was included as a suggested solution in such MKT items in 2001 (Hill, 2007). In 2002, item
developers in the LMT project were required to include “I’m not sure” on all content knowledge
items that were piloted in the area of number concepts and operations. To avoid variation in item
parameters, “I’m not sure” was also included in the 2004 forms that were later translated and
adapted for use in Norway (Fauskanger et al., 2012). “I’m not sure” is always coded as incorrect,
and teachers who are experienced with multiple-choice items would immediately know that this
suggested solution is a distracter (Osterlind, 1997). Some teachers avoid this suggested solution
because they know it will be incorrect; other teachers might select it to avoid guessing or to avoid
giving a wrong answer. In his handbook on development and validation of multiple-choice test
items, Haladyna (2004) recommends that all distracters in multiple-choice items should be
plausible, and it can thus be argued that the suggested solution “I’m not sure” should be avoided.
Given that “I’m not sure” is used in many MKT items, further investigations of teachers’
considerations when faced with such items are called for.
Previous studies indicate that teachers select this alternative solution for different reasons
(Fauskanger & Mosvold, 2014; Haladyna, 2004), and a teacher might select a particular solution
for reasons other than those anticipated (Orrill et al., 2015). In this study, we aim at digging deeper
into this and investigate possible complexities of including “I’m not sure” as a distracter in MKT
items. We analyse data from a collective case study in our endeavour to approach the following
research question:
Why do mathematics teachers select “I'm not sure” when responding to MKT items?
If teachers select “I’m not sure” due to lack of knowledge or uncertainty, the inclusion of this
suggested solution is unproblematic. If, however, teachers select this alternative solution in MKT
multiple-choice items although their responses to other items indicate that they have strong
knowledge, there might be a potential source of misreport in the measures. In answering this
research question, two specific challenges that have been highlighted by the developers of MKT
are addressed: the items might be answered using different types of knowledge such as
procedural knowledge or conceptual knowledge (Ball et al., 2008) and a high MKT score might
be obtained by utilising procedural knowledge only (Fauskanger, 2015; Hill, Umland & Litke,
2012).
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We analyse teachers’ responses to a selection of multiple-choice MKT items and compare this
with their written long responses concerning the content of the same items and their utterances
in group discussions. Adding group discussions is based on recommendations from Adler and
Patahuddin (2012) who report that the MKT items “have much potential in provoking teachers’
talk and their mathematical reasoning in relation to practice-based scenarios; and exploring with
teachers a range of connected knowledge related to the teaching of a particular concept or topic”
(p. 17).
Directed content analysis is applied in our analysis of transcripts from group discussions as
well as the teachers’ long responses (Fauskanger & Mosvold, 2015a; Hsieh & Shannon, 2005).
Tchoshanov’s (2011) framework of cognitive types of teachers’ content knowledge is used as an
analytical lens. We focus in particular on three teachers’ written long responses and discussions
in group interviews regarding MKT items where they selected the alternative “I’m not sure”.
Cross case displays were constructed from teachers’ long responses and their multiple-choice
response supplemented with data from the interviews.
Measuring types of mathematics teachers’ knowledge
It is relevant to gain insight into methods used to access and assess different aspects of teachers’
knowledge (e.g. Hill et al., 2007). One reason why this is important is that teacher knowledge is
allegedly imperative to high quality teaching (e.g. Davis & Simmt, 2006). Here, we concentrate in
particular on accessing and assessing MKT, defined by Hill, Rowan and Ball (2005, p. 373) as, “the
mathematical knowledge used to carry out the work of teaching mathematics”. In the LMT
project, substantial resources have been invested in developing and validating sets of multiple-
choice items in order to assess and access teachers’ MKT (e.g. Schilling, Blunk & Hill, 2007). Based
on these efforts, Hill et al. (2004) suggest that the MKT measures can be used to measure growth
in teachers’ knowledge. The teacher knowledge that is measured by the MKT items further relates
to the mathematical quality of instruction (Hill et al., 2008) and—to a certain extent—to student
learning (Hill et al., 2005; Hill et al., 2012). Inspired by these promising results, researchers have
used the measures extensively in the US, and MKT measures have also been adapted and used
outside the US (Blömeke & Delaney, 2012). Researchers have discussed numerous concerns about
adapting the measures; one concern relates to the challenge of writing clear stems and distracters
(Orrill et al., 2015) and refers to what types of knowledge the measures access (cf. Fauskanger,
2015).
Multiple-choice items
There are some noticeable advantages of multiple-choice items. One advantage is that responses
from multiple-choice items are expeditiously analysed, and they can be used at scale. Developing
multiple-choice items that measure something beyond procedural skills, however, is both tedious
and demanding (e.g. Haladyna, 2004; Osterlind, 1997). When discussing measurement of teacher
knowledge, researchers have argued that use of multiple-choice items might lead to trivialisation
of the complexities of teaching and thus threaten validity (Beswick, Callingham & Watson, 2012;
Haertel, 2004). Even when the target construct is clear, items can measure an unintended
construct, and a teacher can choose a correct or an incorrect answer for reasons other than those
anticipated (Orrill et al., 2015). In his critical discussion of the MKT measures, Schoenfeld (2007)
suggests that MKT items could be measuring something other than they were supposed to. He
adds that the multiple-choice format might complicate the content for the test takers, and
confirming evidence of this was found in a Norwegian context (Fauskanger, Mosvold, Bjuland &
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Jakobsen, 2011). A supplementary aspect of Schoenfeld’s (2007) criticism is that the items
allegedly measure a type of knowledge that is more procedural than intended, and more recent
studies in Norway support this criticism (Fauskanger & Mosvold, 2015b; Fauskanger, 2015).
A standard multiple-choice item consists of two parts. The problem is presented in the first
part of the item, which is often referred to as the item stem. In MKT items, the stem typically
presents a mathematical problem situated in a pedagogical context. In their analyses of the work
of teaching mathematics, the Michigan researchers have sought to decompose the work of
teaching into mathematical tasks of teaching. Hoover, Mosvold and Fauskanger (2014, p. 13)
disclose that, in the MKT items, “mathematical tasks of teaching serve as a kind of backbone
inextricably linking mathematical knowledge to the work of teaching.” However, producing item
stems linking mathematical knowledge to how the knowledge is used in the tasks of teaching
mathematics is challenging. Carefully selected alternative solutions are presented in the second
part of a multiple-choice item. The list of suggested solutions contains a key and one or more
distracters (these are typically incorrect alternatives). In Figure 1, the stem of the question asked,
“Which of these students would you judge to be using a method that could be used to multiply
any two whole numbers?”, but the distractors are Method A, Method B and Method C with 1, 2
and 3 for the different responses. Based on the stem of the question, initially, it might be
challenging to work out what 1, 2 and 3 meant. Some MKT items differ from more standard
multiple-choice items in that mathematically incorrect alternatives are not always included; the
correct solution might then be: “all of the above” or “none of the above”. The use of such items—
where the correct response is to point to several or no correct solutions—are often discouraged
(Haladyna, 2004). If used, they should at least be used with caution (Osterlind, 1997). Another
way in which MKT items differ from more standard multiple-choice items is the extended use of
the suggested solution “I’m not sure” (Fauskanger & Mosvold, 2014)—first included in items that
were formatted yes/no and later for all number and operations content knowledge items (Hill,
2007).
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Figure 1. MKT testlet including “I’m not sure” as a suggested solution in all three multiple-
choice items (Ball & Hill, 2008, p. 5).
The MKT measures are widely adopted (Blömeke & Delaney, 2012). Still, the process of
translating and adapting items is complex (e.g. Fauskanger et al., 2012), and some particular item
distracters seem to be problematic across countries (Ng et al., 2012). Orrill et al. (2015) have
identified five closely interrelated challenges:
1. creating items with appropriate difficulty levels;
2. creating items for the target constructs;
3. using precise language;
4. incorporating pedagogical concerns appropriately; and
5. writing clear stems and distracters (p. 17).
In this article, the fifth challenge is addressed in order to dig deeper into the vagaries of analysing
responses to the MKT multiple-choice items.
Types of teacher knowledge
Different approaches have been made to distinguish between categories of teachers’ knowledge
and understanding of the mathematical content; Skemp’s (1976) categorisation of instrumental
and relational understanding is a classic example. Rote memorisation of algorithms for two-digit
multiplication (Fig. 1) is an example of instrumental understanding, whereas relational
understanding encompasses a deep, conceptual understanding of novel strategies for multi-digit
multiplication. Skemp’s work—along with the work done by the researcher at the University of
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Michigan (e.g. Ball et al., 2008)—was one among several sources of influence when Tchoshanov
(2011) developed his framework of cognitive types of teacher content knowledge. Tchoshanov
differentiated between knowledge of facts and procedures (termed Type 1 knowledge),
knowledge of concepts and connections (termed Type 2 knowledge), and knowledge of models
and generalisations (termed Type 3 knowledge). A person with only procedural knowledge can
multiply 35 by 25 (Fig. 1); this person would multiply the numbers by using an algorithm without
necessarily knowing why it works. A person may, however, also identify the key (i.e., the correct
suggested solution) by employing additional knowledge of concepts and connections (Type 2
knowledge). “Knowing why” is closely related to Tchoshanov’s (2011) Type 2 knowledge. This
type of knowledge is necessary to respond to an item on whether or not a selection of algorithms
for two-digit multiplication could be used to multiply any two whole numbers (as in the testlet
in Figure 1). Type 3 knowledge would relate to the generalised algorithm for multi-digit
multiplication. In his studies, Tchoshanov (2011) did not find strong support for a hypothesis
about the effect of Type 3 knowledge on the quality of instruction and student achievement. The
MKT items included in our study (table 1) do not focus on models or generalisations, and we
have therefore not focused on Type 3 knowledge in the present study.
Skemp (1976) argued that students cannot develop relational understanding from
instrumental teaching. Based on empirical evidence, Tchoshanov (2011) concludes that Type 2
knowledge could be a good predictor of teaching that will have a positive impact on students’
achievement. This conclusion highlights the importance of investigating whether or not measures
of teacher knowledge access Type 2 knowledge (Fauskanger, 2015). For this reason, the
importance of exploring cognitive types of teacher knowledge, accessed by measures of teachers’
MKT as well as possible challenges regarding the use of multiple-choice items to investigate
something as complex as teachers’ MKT, becomes evident. It is relevant to carefully investigate
these challenges (Orrill et al., 2015), and in particular the challenges related to writing stems and
distracters revealed from extensive use of “I’m not sure” as a suggested solution.
Methodology
According to the official coding manuals from the LMT project, the suggested solution “I’m not
sure” should be coded as incorrect. An underlying hypothesis is that teachers who select this
response do not have the MKT necessary to identify the key; they select “I’m not sure” to avoid
giving the wrong answer based on guesswork (Hill, 2007). In order to learn more about why
teachers still select “I’m not sure” in multiple-choice items (e.g. Haladyna, 2004), we compare
each teacher’s responses to the multiple-choice items with the types of knowledge (Tchoshanov,
2011) that can be identified from their written (long) and oral responses to items where they select
the answer “I’m not sure” in the multiple-choice item. The intention is not to compare different
teachers’ MKT, but to compare and contrast the different types of responses each teacher gives to
items. The underlying hypothesis is that teachers’ MKT should be the same regardless of whether
it is accessed by multiple-choice items, written reflections or teachers’ individual voices in group
discussions.
Participants
We applied a case-oriented approach in this study (Silverman, 2006). We considered the case to
be an entity and first looked for configurations and characteristics within the case before we
searched for similarities and patterns across cases. Based on a previous study (Fauskanger &
Mosvold, 2014), and for the purpose of this paper, three in-service teachers were selected as
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cases—from a larger sample of 30 teachers (Fauskanger, 2015)—to investigate the topic under
investigation. The three teachers, who have been assigned the pseudonyms Laura, Ola and Jane,
were participants in a professional development course. They were all experienced mathematics
teachers with more than ten years of experience. The course had 38 participants altogether—all
in-service teachers—and 30 teachers submitted multiple-choice responses to 28 MKT items
(including the three items in the testlet in Fig. 1). Number concepts and operations was the
common theme for all 28 items, and 18 of the items included “I’m not sure” as a suggested
solution (see e.g. Fauskanger & Mosvold, 2014). Fifteen teachers selected this alternative solution
in one, two or three items each (see table 1).

Table 1. Teachers responding “I’m not sure” (Adjusted table from Fauskanger & Mosvold, 2014, p. 47).
Item 1b 1c 1d 5 6a* 6b* 6c* 7d 9d
number
Number 2 2 3 1 2 1 5 1 5
of
teachers
Names Are Are Are, Harald Jane Jane Jane, Frøya Ada,
(pseudo- and and Laura and Ragna, Sara,
nyms) Laura Laura and Ragna Nina, Pia,
Mons Ola Erna
and and
Jan Inge
*6a, 6b and 6c are the multiple-choice items in the testlet presented in Figure 1.
In addition to submitting their multiple-choice responses to the items, the teachers also agreed to
submit long responses related to each item and to discuss the items in groups afterwards. The
teachers were grouped by the grade level they taught. In this article the focus is not on differences
between grade level, formal education or experience; an investigation of such differences is
reported elsewhere (Fauskanger, 2013). The questions prompting long responses were developed
to tap into teachers’ Type 1 and Type 2 knowledge (Tchoshanov, 2011) and varied across the 28
items (see Fauskanger, 2015). These associated open-ended questions were developed by our
group of fellow researchers based on extensive analysis of data from research on accessing and
assessing aspects of mathematics teachers’ knowledge. In MKT, knowledge is defined “in broad
terms” (Ball et al., 2008, p. 399). Consequently, questions were also developed to tap into the type
of knowledge teachers emphasise in connection with their teaching. The questions were
associated with the content of each MKT item specifically and therefore differed from item to
item. The open-ended questions first invited teachers to reflect upon the knowledge
demonstrated by students (e.g. when using a particular algorithm, see Fig. 1). Second, the
questions invited teachers to reflect upon what the students might need to learn as a possible next
step. Third, the teachers were invited to connect the content of the MKT items to their own
teaching practice and to discuss whether or not they would encourage their students to use, for
example, a particular algorithm. Finally, the teachers were asked to provide in-depth reflections
about what knowledge they believe is important for their teaching of mathematics, and whether
or not the 28 MKT items represented important knowledge for them as teachers. The teachers
were asked to justify all their responses.
Efforts have earlier been made to validate the open-ended questions for use in this context
(e.g. Fauskanger & Mosvold, 2012, 2014). In a previous publication, we have reported from
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analysis of multiple-choice responses and long responses for all the teachers in general
(Fauskanger, 2015), and for those who selected “I’m not sure” as a response to one or several
particular items in particular (Fauskanger & Mosvold, 2014). The results from this latter study
indicated that three groups of teachers could be distinguished between: 1) teachers who were not
sure about the content in a particular MKT item, 2) teachers whose long responses indicated Type
1 knowledge, and 3) teachers whose long responses indicated Type 2 knowledge (see table 2, first
row). Laura, Ola and Jane all selected “I’m not sure” in one or more items, but their long responses
indicated that their understanding differed. We consider these three teachers to be representative
of the teachers in their respective groups, and in our attempt to further investigate the rationale
for selecting “I’m not sure” as a response to MKT items, we therefore selected these three teachers
as contrasting cases when analysing group discussions.
Data collection and analyses
Based on analyses of teachers’ written responses (Fauskanger, 2015), group discussions were
implemented. Adding group discussions is based on recommendations from Adler and
Patahuddin (2012) to use MKT items to provoke teachers’ talk and their mathematical reasoning.
Furthermore, Orrill et al. (2015) recommend interviews as a critical approach to meeting the
challenges of multiple-choice item development in general, and to meeting the challenge of
writing clear stems and distracters in particular. The group discussions were based on the same
28 MKT items (Adler & Patahuddin, 2012). The questions forming the point of departure for the
discussions were developed to facilitate more in-depth discussions of the content of 28 MKT
items. Only discussions related to the 18 items including “I’m not sure” as a suggested solution
are analysed for the purpose of this paper.
The unit of analysis is the individual teachers’ multiple-choice responses, their individual
long responses and their individual utterances in group discussions. The group discussions were
recorded and transcribed. We have applied an iterative strategy weaving back and forth between
the empirical material and theories (Alvesson & Karreman, 2011). For the analyses we used
Tchoshanov’s (2011) cognitive types of teacher knowledge (Type 1 and Type 2 knowledge) as
analytic framework. We considered Tchoshanov’s framework to be relevant for this study, since
it builds upon Shulman’s (1986) categorisation of teacher knowledge and also relates to the MKT
framework (Fauskanger, 2015). Tchoshanov’s types were used as categories, and we developed
related codes that were used for coding of teachers’ written long responses and transcripts of
their individual utterances from group discussions. Excerpts from teachers’ written and oral
responses reflecting memorisation of facts or rules, procedural computations or other aspects
related to Type 1 knowledge were coded as “Type 1”. Excerpts reflecting understanding of
concepts and connection between them, multiple solutions to non-routine problems or other
aspects related to Type 2 knowledge were coded as “Type 2”. A third code, low/no knowledge,
was used to code excerpts where teachers explicitly wrote or said that they did not know the
content of the item(s), or excerpts revealing insufficient knowledge. In order to increase the
reliability of the coding, two researchers coded the complete data set independently. In the few
instances where there was a mismatch between our initial coding, we discussed and reached
agreement. Since this was the case only in very few instances, we did not find it necessary to use
any inter-rater reliability method other than individual coding followed by discussion.
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Results and discussion
When we analysed the connection between teachers’ multiple-choice responses and their long
responses according to Tchoshanov’s (2011) framework, three groups of teachers emerged (see
table 2, first row). When analysing the teachers’ responses from group discussions, the same three
groups of teachers could still be distinguished between (see table 2, second row). With one
exception (Jan), the teachers remained in the same groups across written responses and group
discussions.

Table 2. Teachers grouped according to their individual long and oral responses.
Group Group 1: not sure Group 2: Type 1 Group 3: Type 2
knowledge knowledge
Long responses Erna*, Frøya, Jane, Pia, Mons, Harald, Ola Sara, Inge, Ragna and
Jan, Ada and Nina and Are Laura
Oral discussion Frøya, Jane, Ada Jan, Pia, Mons, Harald, Sara, Inge, Ragna and
and Nina Ola and Are Laura
*Not present in the group discussions.
When analysing data from the group discussions, we found it difficult to distinguish between all
the teachers in groups 1 and 2. This becomes apparent from our discussions of the three selected
cases below. In the following, we present results from our analysis of data—multiple-choice
responses, written long responses as well as oral responses in group discussions—from three
teachers: Jane, Ola and Laura.
The case of Jane
Jane’s long responses indicated insecurity related to the content in focus, and she selected “I’m
not sure” as her response to all three items in the testlet in Figure 1 (6a, 6b and 6c in our form).
When asked how she would approach students who use methods like A, B and C, Jane wrote: “It
is difficult to know when you do not understand the methods [the students have] used.” This
long response—along with the other long responses written by Jane—thus seems to support the
hypothesis that the selection of “I’m not sure” implies lack of knowledge or insecurity. Coding
her responses to the multiple-choice items in this testlet as incorrect thus seems reasonable. It can
also be argued that the inclusion of “I’m not sure” has reduced the possibility of guessing with
teachers like Jane, and that was the original intention of including this suggested solution in the
MKT items (Hill, 2007).
In the group interview, Jane explains that she wants to teach an algorithm that she is more
comfortable with herself:
94. Jane: There is a point to explaining that your own way [of calculating it] is all
right, in a way. I have used it for calculating for years, so it is natural to
me. But, then again, we are different. Some like this and some like that.
And it is similar to subtraction, when you borrow 10, if you say for instance
15 minus, or if you say three minus and then add the five. We have shown
both ways, and then it is up to them [the students] what they… Then we
have said that it relates to what they (...) some like this and some like that,
and≈ [sign used to indicate that one interviewee is interrupted by another]
95. Ragna: ≈Open for all [to choose]≈
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96. Inga: ≈And then, many [students] have the parents show it to them in a different
way than the one they have been taught. Yes, that is difficult.
97. Interviewer: Do you experience that the parents have a common [algorithm]?
98. Jane: No. I haven’t asked about that, but I know that many [parents] help their
children setting it up, and with subtraction, it is quite similar with
borrowing (...)
99. Interviewer: Anything else you want to say in relation to the items that were concerning
different algorithms?
100. Jane: No, and we have been taught that there isn’t only one standard algorithm
(laughter).
As displayed in this excerpt, the reason why Jane selected “I’m not sure” seems to be that she is
insecure about the mathematical content, and she wants to teach the students an algorithm with
which she is familiar herself (94). The excerpt thus indicates that Jane holds Type 1 knowledge
(Tchoshanov, 2011) related to multi-digit multiplication, at least related to the algorithms present
in this testlet (Fig. 1). When seen in relation to Jane’s long responses, the excerpts from the group
discussion in which Jane participated could also be interpreted as “not sure” based on her writing
that she does not understand the methods used by the students (see Fig. 1). Jane reveals, however,
that the professional development course has made her aware that multiple algorithms exist, and
she seems to agree that different algorithms might be useful for different students (100).
Based on analyses of Jane’s written and oral responses, her MKT multi-digit multiplication
seems to be somewhere between low and Type 1 (see table 2). It is difficult to distinguish between
these, however, indicated by the arrow between the two cells in table 2. This is also true for other
teachers who were placed in Group 1 (table 2) based on analyses of their written responses. Jan
is one example, and he explicitly said in the group discussion that his view on novel algorithms
for multi-digit multiplication had changed.
The case of Ola
Ola is one of five teachers who selected “I’m not sure” in one of the 18 multiple-choice items that
included this suggested solution. His long responses indicate that he selected “I’m not sure”
based on procedural understanding of the content of the particular item (see table 2). In his long
response related to the testlet presented in Figure 1, Ola writes that his students mainly “learn
only one method by rote without looking for connections.” When faced with students who use
the methods A, B and C in Figure 1, he writes that he would teach them “the standard algorithm
for multiplication.” He will not encourage his students to use the methods A, B and C, because it
will be “too much work.” Ola teaches in the lower secondary grades, and when writing about
algorithms for multi-digit subtraction and multiplication he has not made use of more than one
algorithm because “most students have learned the same algorithm I learned myself, 40 years
ago, and I don’t know if I have experienced anyone not using this when subtracting. There has
been one or two examples where students have used a method of multiplication that was
unknown to me.” Ola also explicitly writes that his knowledge related to algorithms is
procedural. At the end of the first semester, he writes “After 3 months in this course, I consider it
[knowing more than one algorithm for multi-digit arithmetic] more useful than previously
thought. I have previously considered the algorithms for the four mathematical operations only
as learned, automatic procedures. I have thought of our place value system mainly as ‘innate’
knowledge.” This excerpt indicates that, during the course, he might have developed a type of
knowledge that exceeds Type 1 knowledge (Tchoshanov, 2011), but these written reflections
alone are not sufficient to confirm his increased knowledge.
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